Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses Statistics.

Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses

Statistics

McClave, Statistics, 11th ed. Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses

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Where We’ve Been

Calculated point estimators of population parameters

Used the sampling distribution of a statistic to assess the reliability of an estimate through a confidence interval


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Where We’re Going

Test a specific value of a population parameter

Measure the reliability of the test

8.1: The Elements of a Test of Hypotheses


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Confidence IntervalWhere on the number line do the data point us?(No prior idea about the value of the parameter.)

Hypothesis TestDo the data point us to this particular value?(We have a value in mind from the outset.)

µ? µ?

µ0?



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Null Hypothesis: H0

•This will be supported unless the data provide evidence that it is false• The status quo Alternative Hypothesis: Ha

•This will be supported if the data provide sufficient evidence that it is true• The research hypothesis


If the test statistic has a high probability when H0 is true, then H0 is not rejected.

If the test statistic has a (very) low probability when H0 is true, then H0 is rejected.


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Reality ↓ / Test Result → Do not reject H0 Reject H0

H0 is true Correct!Type I Error: rejecting a true null hypothesisP(Type I error) =

H0 is falseType II Error: not rejecting a false null hypothesisP(Type II error) =

Correct!


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Note: Null hypotheses are either rejected, or else there is insufficient evidence to reject them. (I.e., we don’t accept null hypotheses.)


• Null hypothesis (H0): A theory about the values of one or more parameters

• Ex.: H0: µ = µ0 (a specified value for µ)

• Alternative hypothesis (Ha): Contradicts the null hypothesis

• Ex.: H0: µ ≠ µ0

• Test Statistic: The sample statistic to be used to test the hypothesis• Rejection region: The values for the test statistic which lead to rejection of

the null hypothesis• Assumptions: Clear statements about any assumptions concerning the

target population• Experiment and calculation of test statistic: The appropriate calculation for

the test based on the sample data• Conclusion: Reject the null hypothesis (with possible Type I error) or do

not reject it (with possible Type II error)


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Suppose a new interpretation of the rules by soccer referees is expected to increase the number of yellow cards per game. The average number of yellow cards per game had been 4. A sample of 121 matches produced an average of 4.7 yellow cards per game, with a standard deviation of .5 cards. At the 5% significance level, has there been a change in infractions called?


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H0: µ = 4Ha: µ ≠ 4Sample statistic: = 4.7= .05

Assume the sampling distribution is normal.

Test statistic:

Conclusion: z.05 = 1.96. Since z* > z.05 , reject H0.(That is, there do seem to be more yellow cards.)

94.10064.

47.4* 0

xs

xz

8.2: Large-Sample Test of a Hypothesis about a Population Mean


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The null hypothesis isusually stated as an equality …

… even though the alternative hypothesis can be either an equality or an inequality.



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Lower Tailed Upper Tailed Two tailed

= .10 z < - 1.28 z > 1.28 | z | > 1.645

= .05 z < - 1.645 z > 1.645 | z | > 1.96

= .01 z < - 2.33 z > 2.33 | z | > 2.575


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Rejection Regions for Common Values of


One-Tailed Test H0 : µ = µ0

Ha : µ < or > µ0

Test Statistic:

Rejection Region: | z | > z

Two-Tailed Test


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x

xz

0

H0 : µ = µ0

Ha : µ ≠ µ0

Test Statistic:

Rejection Region: | z | > z/2

x

xz

0

Conditions: 1) A random sample is selected from the target population.2) The sample size n is large.


The Economics of Education Review (Vol. 21, 2002) reported a mean salary for males with postgraduate degrees of $61,340, with an estimated standard error (s ) equal to $2,185. We wish to test, at the = .05 level, H0: µ = $60,000.


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The Economics of Education Review (Vol. 21, 2002) reported a mean salary for males with postgraduate degrees of $61,340, with an estimated standard error (s) equal to $2,185. We wish to test, at the = .05 level, H0: µ = $60,000.


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H0 : µ = 60,000

Ha : µ ≠ 60,000

Test Statistic:

Rejection Region: | z | > z.025 = 1.96

613.

185,2

000,60340,61

0

z

z

xz

x

Do not reject H0

8.3:Observed Significance Levels: p - Values


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Suppose z = 2.12. P(z > 2.12) = .0170.

Reject H0 at the = .05 level Do not reject H0 at the = .01 level

But it’s pretty close, isn’t it?



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The observed significance level, or p-value, for a test is the probability of observing the results actually observed (z*) assuming the null hypothesis is true.

The lower this probability, the less likely H0 is true.

)|*( 0HzzP

Let’s go back to the Economics of Education Review report (= $61,340, s = $2,185). This time we’ll test H0: µ = $65,000.


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H0 : µ = 65,000

Ha : µ ≠ 65,000

Test Statistic:

p-value: P( 61,340 |H0 ) = P(|z| > 1.675) = .0475

675.1

185,2

000,65340,61

0

z

z

xz

x



Reporting test results Choose the maximum tolerable value of If the p-value < , reject H0

If the p-value > , do not reject H0


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Some stats packages will only report two-tailed p-values.



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positive is andincludes

negative is andincludes if

2

value- Reported1

negative is andincludes

positive is andincludes if

2

value- Reported

zH

zHpp

zH

zHpp

a

a

a

a

Converting a Two-Tailed p-Value to a One-Tailed p-Value

Some stats packages will only report two-tailed p-values.



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Converting a Two-Tailed p-Value to a One-Tailed p-Value

8.4: Small-Sample Test of a Hypothesis about a Population Mean


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If the sample is small and is unknown, testing hypotheses about µ requires the t-distribution instead of the z-distribution.

ns

xt

/0

One-Tailed Test H0 : µ = µ0

Ha : µ < or > µ0

Test Statistic:

Rejection Region: | t | > t

Two-Tailed Test


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ns

xt

/0

H0 : µ = µ0

Ha : µ ≠ µ0

Test Statistic:

Rejection Region: | t | > t/2

ns

xt

/0

Conditions: 1) A random sample is selected from the target population.2) The population from which the sample is selected is

approximately normal.3) The value of t is based on (n – 1) degrees of freedom



Suppose copiers average 100,000 between paper jams. A salesman claims his are better, and offers to leave 5 units for testing. The average number of copies between jams is 100,987, with a standard deviation of 157. Does his claim seem believable?


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H0 : µ = 100,000

Ha : µ > 100,000

Test Statistic:

p-value: P( 100,987|H0 ) = P(|tdf=4| > 14.06) < .001




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06.145/157

000,100987,100/

0

t

t

ns

xt

H0 : µ = 100,000

Ha : µ > 100,000

Test Statistic:

p-value: P( 100,987|H0 ) = P(|tdf=4| > 14.06) < .001




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06.145/157

000,100987,100/

0

t

t

ns

xt

Reject the null hypothesis based on the very low probability of seeing the observed results if the null were true.So, the claim does seem plausible.

One-Tailed Test H0 : p = p0

Ha : p < or > p0

Test Statistic:

Rejection Region: | z | > z

Two-Tailed Test


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p

ppz

ˆ

0ˆ

H0 : p = p0

Ha : p ≠ p0

Test Statistic:

Rejection Region: | z | > z/2

p

ppz

ˆ

0ˆ

Conditions: 1) A random sample is selected from a binomial population. 2) The sample size n is large (i.e., np0 and nq0 are both 15).

8.5: Large-Sample Test of a Hypothesis about a Population Proportion

p0 = hypothesized value of p, , and q0 = 1 - p0 n

qpp

00ˆ


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Rope designed for use in the theatre must withstand unusual stresses. Assume a brand of 3” three-strand rope is expected to have a breaking strength of 1400 lbs. A vendor receives a shipment of rope and needs to (destructively) test it.

The vendor will reject any shipment which cannot pass a 1% defect test (that’s harsh, but so is falling scenery during an aria). 1500 sections of rope are tested, with 20 pieces failing the test. At the = .01 level, should the shipment be rejected?


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The vendor will reject any shipment that cannot pass a 1% defects test . 1500 sections of rope are tested, with 20 pieces failing the test. At the = .01 level, should the shipment be rejected?

H0: p = .01

Ha: p > .01

Rejection region: |z| > 2.236

Test statistic:

14.1

1500/)987)(.013(.

01.013.

ˆ

ˆ

0

z

z

ppz

p


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The vendor will reject any shipment that cannot pass a 1% defects test . 1500 sections of rope are tested, with 20 pieces failing the test. At the = .01 level, should the shipment be rejected?

H0: p = .01

Ha: p > .01

Rejection region: |z| > 2.236

Test statistic:

14.1

1500/)987)(.013(.

01.013.

ˆ

ˆ

0

z

z

ppz

p

There is insufficient evidence to reject the null hypothesis based on the sample results.

There is insufficient evidence to reject the null hypothesis based on the sample results.

8.6: Calculating Type II Error Probabilities: More about

To calculate P(Type II), or , …1. Calculate the value(s) of that divide the “do not reject” region from the “reject” region(s).

Upper-tailed test:

Lower-tailed test:

Two-tailed test:


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n

szzx

n

szzx

n

szzx

n

szzx

xU

xL

x

x

2/02/00

2/02/00

000

000


To calculate P(Type II), or , …1. Calculate the value(s) of that divide the “do not reject” region from the “reject” region(s).

2. Calculate the z-value of 0 assuming the alternative hypothesis mean is the true µ:

The probability of getting this z-value is .


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x

axz

0


The power of a test is the probability that the test will correctly lead to the rejection of the null hypothesis for a particular value of µ in the alternative hypothesis. The power of a test is calculated as (1 - ).


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The Economics of Education Review (Vol. 21, 2002) reported a mean salary for males with postgraduate degrees of $61,340, with an estimated standard error (s) equal to $2,185. We wish to test, at the = .05 level, H0: µ = $60,000.

H0 : µ = 60,000

Ha : µ ≠ 60,000

Test Statistic: z = .613;

z=.025 = 1.96

We did not reject this null hypothesis earlier, but what if the true mean were $62,000?



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The Economics of Education Review (Vol. 21, 2002) reported a mean salary for males with postgraduate degrees of $61,340, with s equal to $2,185.

We did not reject this null hypothesis earlier, but what if the true mean were $62,000?

3821.30.0

2185

000,62340,61

)(

zP

zP

IITypeP

The power of this test is1 - .3821 = .6179


For fixed n and , the value of decreases and the power increases as the distance between µ0 and µa increases.

For fixed n, µ0 and µa, the value of increases and the power decreases as the value of is decreased.

For fixed , µ0 and µa, the value of decreases and the power increases as n is increased.


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8.7: Tests of Hypotheses about a Population Variance


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2

22 )1(

sn

The chi-square distribution is really a family of distributions, depending on the number of degrees of freedom.

But, the population must be normally distributed for the hypothesis tests on 2 (or ) to be reliable!


One-Tailed Test

Test statistic:

Rejection region:


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2221

2

20

22

20

220

2

20

20

)1(

:

:

sn

H

H

a

Two-Tailed Test

Test statistic:

Rejection region:2

2/22

2/12

20

22

20

2

20

20

)1(

:

:

or

sn

H

H

a


Conditions Required for a Valid

Large- Sample Hypothesis Test for 2

1. A random sample is selected from the target population.

2. The population from which the sample is selected is approximately normal.


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Earlier, we considered the average number of copies between jams for a brand of copiers. The salesman also claims his copiers are more predictable, in that the standard deviation of jams is 125. In the sample of 5 copiers, that sample standard deviation was 157. Does his claim seem believable, at the = .10 level?


Two-Tailed Test

Test statistic:

Rejection criterion:


46

48773.931.6

125

)157)(15(

125:

125:

205.

2

2

22

2

20

aH

H

Earlier, we considered the average number of copies between jams for a brand of copiers. The salesman also claims his copiers are more predictable, in that the standard deviation of jams is 125. In the sample of 5 copiers, that sample standard deviation was 157. Does his claim seem believable, at the = .10 level?


Two-Tailed Test

Test statistic:

Rejection criterion:


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48773.931.6

125

)157)(15(

125:

125:

205.

2

2

22

2

20

aH

H

Earlier, we considered the average number of copies between jams for a brand of copiers. The salesman also claims his copiers are more reliable, in that the standard deviation of jams is 125. In the sample of 5 copiers, that sample standard deviation was 157. Does his claim seem believable, at the = .10 level?

Do not reject the null

hypothesis.

Chapter 8: Inferences Based on a Single Sample: Tests of Hypotheses Statistics.

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